Classification of Algebraic Function Fields with Class Number One

TL;DR

This paper proves that there are exactly eight algebraic function fields over finite fields with class number one, providing a complete classification of such fields.

Contribution

It establishes a complete classification of algebraic function fields with class number one over finite fields, identifying all such fields.

Findings

Exactly eight such function fields exist over finite fields.

The classification is complete and up to isomorphism.

Provides foundational knowledge for further algebraic function field research.

Abstract

In this paper we prove that there are exactly eight function fields, up to isomorphism, over finite fields with class number one.